Kramers-Kronig constrained modeling of soft x-ray reflectivity spectra: Obtaining depth resolution of electronic and chemical structure

Abstract

Resonant x-ray scattering is a powerful technique for the study of electronic structure at the nanoscale. In common practice, the optical properties of the constituent components of a material must be known prior to modeling of the scattered intensity. We present a means of refining electronic structure, in the form of optical properties, simultaneous to physical structure, in a Kramers-Kronig consistent manner. This approach constitutes a sensitive and powerful extension of resonant x-ray scattering to materials where the optical properties are not sufficiently well known. The application of this approach to specular reflectivity from a single crystal of SrTiO3 is presented as an example case, wherein we find evidence for both a non-resonant surface contaminant layer and a modified SrTiO3 surface region. Extrapolating from this study we comment on the potential utility of this approach to resonant scattering studies in general.2

Magnetic reflectometry of heterostructures

“…The reflected intensity follows the Fresnel equations in which δ and β of the substrate are the only unknowns. By Kramers-Kronig consistent fitting of the fixed q z -scans the quantitative correct optical constants can be retrieved [88][89][90][91]. This approach is still new and under development.…”

Spectral curve fitting of dielectric constants

“…17 The idea is to find a set of triangular basis functionsε j (v) anchored at ν j so that the imaginary part of the dielectric constant can be expressed as a linear combination of these basis functions. 6,27 One such triangular function is: 17…”

“…As an intrinsic property of the material, 1 optical constants contain information on the composition of material, 2-4 its anisotropy, [5][6][7] and its optical activity. 8 The optical properties of a material can be inferred from experimental spectra from X-ray frequencies through to the Far-Infrared/Terahertz regions, in either transmittance or absorbance or reflectance mode.…”

Molecular anisotropy effects in carbonK-edge scattering: Depolarized diffuse scattering and optical anisotropy

“…This analysis differs from those used in previous studies of anisotropic carbon edge reflectivity [16][17][18] in that it obtains absolute optical properties directly from measured reflectivity spectra. We use a maximum entropy refinement algorithm that ensures Kramers-Kronig consistency to obtain the complex resonant monomer scattering factors 〈 (ℎ )〉 / = 〈 1 〉 / + 〈 2 〉 / that best fit the data [40]. Brackets indicate values spatially averaged over the x-ray coherence volume and the larger illuminated volume of the sample.…”

“…From the analysis of reflectivity data at one angle presented here, it is impossible 24 to draw conclusions about possible inhomogeneous optical anisotropy in depth. Analysis of data measured at several incidence angles does facilitate such depth-dependent modeling of changing resonant optical properties [17,40,71]. As polymer chains and interactions deviate further from random coil behavior, such as in conjugated polymers, we can expect increasing optical anisotropy effects in homopolymer films.…”